| Variable | Description | Type |
|---|---|---|
| diabetes_dx | Diabetes diagnosis (1 = Yes, 0 = No) based on medical questionnaire. | Categorical |
| age | Age of participant in years. | Continuous |
| bmi | Body Mass Index (BMI) in kilograms per square meter (kg/m²), calculated from measured height and weight. | Continuous |
| sex | Sex of participant (Male or Female). | Categorical |
| race | Race/Ethnicity (e.g., Non-Hispanic White, Non-Hispanic Black, Mexican American, etc.). | Categorical |
| WTMEC2YR | Examination sample weight for MEC (Mobile Examination Center) participants. | Weight |
| SDMVPSU | Primary Sampling Unit (PSU) used for variance estimation in complex survey design. | Design |
| SDMVSTRA | Stratum variable used to define strata for complex survey design. | Design |
| age_c | Age variable centered and standardized (z-score). | Continuous |
| bmi_c | BMI variable centered and standardized (z-score). | Continuous |
| wt_norm | Normalized survey weight (WTMEC2YR divided by its mean, for model weighting). | Weight |
Bayesian Logistic Regression - Application in Probability Prediction of disease (Diabetes)
CapStone Project_2025
Slides: slides.html ( Go to slides.qmd to edit)
Introduction
Diabetes mellitus (DM) is a major public health concern closely associated with factors such as obesity, age, race, and gender. Identifying these associated risk factors is essential for targeted interventions D’Angelo and Ran (2025). Logistic Regression (traditional) that estimates the association between risk factors and outcomes is insufficient in analyzing the complex healthcare data (DNA sequences, imaging, patient-reported outcomes, electronic health records (EHRs), longitudinal health measurements, diagnoses, and treatments. Zeger et al. (2020). Classical maximum likelihood estimation (MLE) yields unstable results in samples that are small, have missing data, or presents quasi- and complete separation.
Bayesian hierarchical models using Markov Chain Monte Carlo (MCMC) allow analysis of multivariate longitudinal healthcare data with repeated measures within individuals and individuals nested in a population. By integrating prior knowledge and including exogenous (e.g., age, clinical history) and endogenous (e.g., current treatment) covariates, Bayesian models provide posterior distributions and risk predictions for conditions such as pneumonia, prostate cancer, and mental disorders. Parametric assumptions remain a limitation of these models.
In Bayesian inference Chatzimichail and Hatjimihail (2023), Bayesian inference has shown that parametric models (with fixed parameters) often underperform compared to nonparametric models, which do not assume a prior distribution. Posterior probabilities from Bayesian approaches improve disease classification and better capture heterogeneity in skewed, bimodal, or multimodal data distributions. Bayesian nonparametric models are flexible and robust, integrating multiple diagnostic tests and priors to enhance accuracy and precision, though reliance on prior information and restricted access to resources can limit applicability. Combining Bayesian methods with other statistical or computational approaches helps address systemic biases, incomplete data, and non-representative datasets.
The Bayesian framework described by R. van de Schoot et al. (2021) highlights the role of priors, data modeling, inference, posterior sampling, variational inference, and variable selection.Proper variable selection mitigates multicollinearity, overfitting, and limited sampling, improving predictive performance. Priors can be informative, weakly informative, or diffuse, and can be elicited from expert opinion, generic knowledge, or data-driven methods. Sensitivity analysis evaluates the alignment of priors with likelihoods, while MCMC simulations (e.g., brms, blavaan in R) empirically estimate posterior distributions. Spatial and temporal Bayesian models have applications in large-scale cancer genomics, identifying molecular interactions, mutational signatures, patient stratification, and cancer evolution, though temporal autocorrelation and subjective prior elicitation can be limiting.
Bayesian normal linear regression has been applied in metrology for instrument calibration using conjugate Normal–Inverse-Gamma priors Klauenberg et al. (2015). Hierarchical priors add flexibility by modeling uncertainty across multiple levels, improving robustness and interpretability. Bayesian hierarchical/meta-analytic linear regression incorporates both exchangeable and unexchangeable prior information, addressing multiple testing challenges, small sample sizes, and complex relationships among regression parameters across studies Leeuw and Klugkist (2012)
A sequential clinical reasoning model Liu et al. (2013) Sequential clinical reasoning models demonstrate screening by adding predictors stepwise: (1) demographics, (2) metabolic components, and (3) conventional risk factors, incorporating priors and mimicking clinical evaluation. This approach captures ecological heterogeneity and improves baseline risk estimation, though interactions between predictors and external cross-validation remain limitations.
Bayesian multiple imputation with logistic regression addresses missing data in clinical research Austin et al. (2021) in clinical research by classifying missing values (e.g., patient refusal, loss to follow-up, mechanical errors) as MAR, MNAR, or MCAR. Multiple imputation generates plausible values across datasets and pools results for reliable classification of patient health status and mortality.
Aims
The present study aims performs Bayesian logistic regression to predict diabetes status and evaluate the associations between diabetes and predictors (body mass index (BMI), age (≥20 years), gender, and race). The study anakyzes a retrospective dataset (2013–2014 NHANES survey data). It is based on a complex sampling design, characterized by stratification, clustering, and oversampling of specific population subgroups, rather than uniform random sampling. A Bayesian analytical approach addresses challenges posed by dataset anomalies such as missing data, complete case analysis, and separation that limit the efficiency and reliability of traditional logistic regression in predicting health outcomes.
Method
Bayesian Logistic Regression
The study employs Bayesian logistic regression to estimate associations between predictors and outcome probabilities.
The Bayesian framework integrates prior information with observed data to generate posterior distributions, allowing direct probabilistic interpretation of parameters.
This approach provides flexibility in model specification, accounts for parameter uncertainty, and produces credible intervals that fully reflect uncertainty in the estimates.
Unlike traditional frequentist methods, the Bayesian method enables inference through posterior probabilities, supporting more nuanced decision-making and interpretation.
Model Structure
Bayesian logistic regression
Bayesian logistic regression is a probabilistic modeling framework used to estimate the relationship between one or more predictors (continuous or categorical) and a binary outcome (e.g., presence/absence of disease).
It extends classical logistic regression by combining it with Bayesian inference, treating model parameters as random variables with probability distributions rather than fixed point estimates Leeuw and Klugkist (2012) and Klauenberg et al. (2015)
The logistic model relates the probability of an outcome ( Y = 1 ) to a linear combination of predictors through the logit link function:
[ (P(Y = 1)) = _0 + _1 X_1 + _2 X_2 + + _k X_k ]
logit(pi)=β0+j=1∑pβjxij
- p_i: the probability of the event (e.g., having diabetes) for individual i.
- “logit”(p_i)=log(p_i/(1-p_i )): the log-odds of the event.
- β_0: the intercept — the log-odds of the event when all predictors x_ij=0.
- β_j: the coefficient for predictor x_j, representing the change in log-odds for a one-unit increase in x_ij, holding other variables constant.
- ∑_(j=1)^p β_j x_ij: the combined linear effect of all predictors.
In the Bayesian framework, the coefficients () are assigned prior distributions, which are updated in light of the observed data to yield posterior distributions.
The Bayesian approach naturally incorporates uncertainty in all model parameters.
It combines prior beliefs with observed data to produce posterior distributions according to Bayes’ theorem: [ ]
Likelihood: is the probability of the observed data given the model parameters (as in classical logistic regression).
Prior: Encodes prior knowledge or beliefs about parameter values before observing the data.
Posterior: is the updated beliefs about parameters after observing the data.
Posterior ⋉ Likelihood * Prior
Prior Specification
Regression coefficients prior: normal(0, 2.5) — providing weakly informative regularization provide gentle regularization, constraining extreme values without overpowering the data R. van de Schoot et al. (2021)
A weakly informative intercept prior: Student’s t-distribution prior, student_t(3, 0, 10) (van de Schoot et al., 2013).
This prior:
- Has 3 degrees of freedom (( = 3 )), producing heavy tails that allow for occasional large effects.
- Is centered at 0 (( = 0 )), reflecting no initial bias toward positive or negative associations.
- Has a scale parameter of 10 (( = 10 )), allowing broad variation in possible coefficient values.
Such priors improve stability in models with small sample sizes, high collinearity, or potential outliers.
Advantages of Bayesian Logistic Regression
- Uncertainty quantification: Produces full posterior distributions instead of single-point estimates.
- Credible intervals: Provide the range within which a parameter lies with a specified probability (e.g., 95%).
- Flexible priors: Allow incorporation of expert knowledge or results from previous studies.
- Probabilistic predictions: Posterior predictive distributions yield direct probabilities for future observations.
- Comprehensive model checking: Posterior predictive checks (PPCs) evaluate how well simulated outcomes reproduce observed data.
Posterior Predictions
- Posterior distributions of the coefficients are used to estimate the probability of the outcome for given predictor values. This enables statements such as:
“Given the predictors, the probability of the outcome lies between X% and Y%.” - Posterior predictions incorporate two sources of uncertainty:
- Parameter uncertainty: Variability in estimated model coefficients.
- Predictive uncertainty: Variability in future outcomes given those parameters.
- In Bayesian analysis, every unknown parameter — such as a regression coefficient, mean, or variance — is treated as a random variable with a probability distribution that expresses uncertainty given the observed data.
Model Evaluation and Diagnostics
- Model quality and convergence are assessed using standard Bayesian diagnostics:
- Posterior inference performed using Markov Chain Monte Carlo (MCMC) sampling via the No-U-Turn Sampler (NUTS), a variant of Hamiltonian Monte Carlo (HMC). Four chains run with sufficient warm-up iterations to ensure convergence. Austin et al. (2021)
- Convergence diagnostics: Markov Chain Monte Carlo (MCMC) performance was evaluated using ( ) (R-hat) and effective sample size (ESS).
- Autocorrelation checks: Ensure independence between successive MCMC draws.
- Posterior predictive checks (PPC): Compare simulated data from posterior distributions to observed outcomes.
- Bayesian R²: Quantified the proportion of variance explained by the predictors, incorporating uncertainty.
Analysis and Results
Statistical Tool
R packages and libraries are used to import data, perform data wrangling and analysis. All computation is implemented in a probabilistic programming environment using the brms interface to Stan.
Data source
- NHANES 2-year data (2013-2014) cross-sectional weighted data Center for Health Statistics (1999) was imported in R
Data pre-processing
- Adult dataset:Three NHANES datasets are merged: demographics (DEMO_H), body measures (BMX_H), and the diabetes questionnaire (DIQ_H) in (.XPT) format, imported via Haven package in R. Variables of interest are selected using the original weighted datasets and ID to create a single adult analytic dataframe.
- All variables were coerced to consistent numeric or factor types prior to merging to ensure atomic columns suitable for survey analysis and modeling.
Data Variables
Response Variable - Binary Type 2 / diagnosed diabetes(excluding gestational diabetes) diabetes_dx created combning - DIQ010 - Doctor told you have diabetes - DIQ050- excluded (a secondary variable describing treatment status (insulin use)). Predictor Variables - Body Mass Index, factor, 4 levels
Covariates - Gender (factor, 2 levels) - Ethnicity (factor, 5 levels) - Age (continuous 20-80 years)
Study Design and Survey-Weighted Analysis
NHANES is a national survey using complex, multistage sampling, including oversampling of certain groups (e.g., minorities, older adults) to ensure adequate representation.
Survey design variables — primary sampling units (SDMVPSU), strata (SDMVSTRA), and examination weights (WTMEC2YR) — were retained to account for this design.
Survey-weighted logistic regression is used to examine the association between diabetes status (binary outcome: diabetes_dx) and predictors including BMI, age, sex, and race/ethnicity.
Sampling weights, strata, and PSU were applied to: - Correct for unequal probabilities of selection and nonresponse. - Produce population-representative estimates. - Ensure valid standard errors and inference for U.S. adults. - Diabetes was derived from NHANES variables (DIQ010, excluding DIQ050) and coded as 0/1 (diabetes_dx).
Covariates included:
Age (continuous, centered age_c, and categorized 20–80 years) BMI (continuous, centered bmi_c, and categorical bmi_cat) Sex (male, female) Race/ethnicity (5 levels)
This approach ensures unbiased prevalence estimates and valid inference for population-level associations while accounting for NHANES’ complex sampling design.
| Step | Description- recoding, categorization, missing data and final data |
|---|---|
| Weighting | Used the survey package to calculate weighted means and standard deviations for all variables. |
| Standardization | Standardized BMI and age variables for analysis. |
| Age Categorization | Recoded into intervals: 20–<30, 30–<40, 40–<50, 50–<60, 60–<70, and 70–80 years. |
| BMI Categorization | Recoded and categorized as: <18.5 (Underweight), 18.5–<25 (Normal), 25–<30 (Overweight), 30–<35 (Obesity I), 35–<40 (Obesity II), ≥40 (Obesity III). |
| Ethnicity Recoding | Recoded as: 1 = Mexican American, 2 = Other Hispanic, 3 = Non-Hispanic White, 4 = Non-Hispanic Black, 5 = Other/Multi. |
| Special Codes | Special codes (e.g., 3, 7) were transformed to NA. These codes are not random and could introduce bias if ignored (MAR or MNAR). |
| Missing Data | Missing values were retained and visualized to assess their pattern and informativeness. |
| Final Dataset | Created a cleaned analytic dataset (adult) using Non-Hispanic White and Male as reference groups for analysis. |
Exploratory Data Analysis (Adult, 20 - 80 years)
- Tabulated below is the Adult dataframe structure, strucutre plot and showcase of the initial readings
- Mean, standard error and variance of the survey weighted data and effective sample size
mean SE
age 47.496 0.3805 mean SE
diabetes_dx 0.089016 0.0048 variance SE
diabetes_dx 4759.9 0.0039Effective sample size for diabetes_dx: 48142 Code
library(dplyr)
library(skimr)
library(knitr)
library(tidyr)
library(purrr)
library(forcats)
library(kableExtra)
str(adult)'data.frame': 5769 obs. of 12 variables:
$ SDMVPSU : num 1 1 1 2 1 1 2 1 2 2 ...
$ SDMVSTRA : num 112 108 109 116 111 114 106 112 112 113 ...
$ WTMEC2YR : num 13481 24472 57193 65542 25345 ...
$ diabetes_dx: num 1 1 1 0 0 0 0 0 0 0 ...
$ bmi : num 26.7 28.6 28.9 19.7 41.7 35.7 NA 26.5 22 20.3 ...
$ age : num 69 54 72 73 56 61 42 56 65 26 ...
$ sex : Factor w/ 2 levels "Male","Female": 1 1 1 2 1 2 1 2 1 2 ...
$ race : Factor w/ 5 levels "NH White","Mexican American",..: 4 1 1 1 2 1 3 1 1 1 ...
$ DIQ050 : num 1 1 1 2 2 2 2 2 2 2 ...
$ age_c : num 1.132 0.278 1.303 1.36 0.392 ...
$ bmi_c : num -0.3359 -0.0703 -0.0283 -1.3144 1.761 ...
$ bmi_cat : Factor w/ 6 levels "<18.5","18.5–<25",..: 3 3 3 2 6 5 NA 3 2 2 ...Code
plot_str(adult)
head(adult) SDMVPSU SDMVSTRA WTMEC2YR diabetes_dx bmi age sex race DIQ050
1 1 112 13481.04 1 26.7 69 Male NH Black 1
2 1 108 24471.77 1 28.6 54 Male NH White 1
3 1 109 57193.29 1 28.9 72 Male NH White 1
4 2 116 65541.87 0 19.7 73 Female NH White 2
5 1 111 25344.99 0 41.7 56 Male Mexican American 2
6 1 114 61758.65 0 35.7 61 Female NH White 2
age_c bmi_c bmi_cat
1 1.1324183 -0.33588609 25–<30
2 0.2783598 -0.07028101 25–<30
3 1.3032300 -0.02834336 25–<30
4 1.3601672 -1.31443114 18.5–<25
5 0.3922343 1.76099614 ≥40
6 0.6769204 0.92224325 35–<40Code
plot_intro(adult, title="Figure 1 (Adult dataset). Structure of variables and missing observations.")
Code
plot_missing(adult, title="Figure 2(Adult dataset). Breakdown of missing observations.")
Bar graph of Adult dataframe (variable structure and missingness)
- 25% of columns are discrete (categorical)
- 75% are continuous, indicating that the dataset primarily contains continuous measurements such as age and BMI.
- 92.7% of rows have complete information for all variables, meaning most participants have fully observed data across predictors and outcomes.
- Number of participants in Adult dataset (n = 5769)
- Age: Participants are fairly evenly distributed across adult age groups, with no sharp skewness.
- Sex: Sample includes a higher proportion of females than males.
- BMI: Most participants have BMI values within the normal to overweight range, with fewer in the obese category.
- Diabetes Status: Plot shows prevalence of Diabetes
- Diabetes by BMI categories: Individuals diagnosed with diabetes tend to have higher BMI values compared to non-diabetics.
- Diabetesby Age Group: The proportion of diabetes increases with advancing age, highlighting age as a strong risk factor.
- Diabetes by Race/Ethnicity: Differences are observed across racial/ethnic groups, with some showing higher prevalence rates than others.
- Diabetes diagnosis by sex across different racial groups. Bars are side by side for each sex, with counts displayed on top
Code
ggplot(adult, aes(x = age)) +
geom_histogram(binwidth = 5, fill = "skyblue", color = "white") +
labs(
title = "Distribution of Age >20 years",
x = "Age (years)",
y = "Count"
) +
theme_minimal()
Code
ggplot(adult, aes(factor(diabetes_dx))) +
geom_bar(fill = "steelblue") +
labs(title="Diabetes Outcome Distribution in >20 years age group", x="diabetes_dx (0=No, 1=Yes)", y="Count")
Code
ggplot(adult, aes(factor(bmi_cat))) +
geom_bar(fill = "steelblue") +
labs(title="Diabetes Outcome Distribution by BMI in >20 years age group", x="bmi_cat")
Code
ggplot(adult, aes(x = factor(diabetes_dx), y = bmi)) +
geom_boxplot(fill = "skyblue") +
labs(
title = "BMI Distribution by Diabetes Diagnosis in >20 years age group",
x = "Diabetes Diagnosis (0 = No, 1 = Yes)",
y = "BMI"
) +
theme_minimal()
Code
# plots for adult data bmi categories and race categories
ggplot(adult, aes(x = factor(race), fill = factor(diabetes_dx))) +
geom_bar(position = "dodge") +
labs(
title = "Diabetes Diagnosis by Race in >20 years age group",
x = "Race/Ethnicity",
y = "Count",
fill = "Diabetes Diagnosis\n(0 = No, 1 = Yes)"
) +
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
Code
ggplot(adult, aes(x = factor(bmi_cat), fill = factor(diabetes_dx))) +
geom_bar(position = "dodge") +
labs(
title = "Diabetes Diagnosis by BMI in >20 years age group",
x = "BMI",
y = "Count",
fill = "Diabetes Diagnosis\n(0 = No, 1 = Yes)"
) +
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
Code
# Example: create your dataset
adult1 <- data.frame(
race = rep(c("NH White","Mexican American","Other Hispanic","NH Black","Other/Multi"), each = 6),
sex = rep(c("Male","Male","Male","Female","Female","Female"), times = 5),
diabetes_dx = rep(c(0,1,NA,0,1,NA), times = 5),
count = c(
1019,119,38,1164,96,36,
304,60,14,329,49,11,
183,26,10,255,25,9,
461,100,19,515,65,17,
351,46,8,393,32,15
)
)
# Clean NA for plotting or convert to "Missing"
adult1$diabetes_dx <- as.character(adult1$diabetes_dx)
adult1$diabetes_dx[is.na(adult1$diabetes_dx)] <- "Missing"
# Plot grouped bar chart
ggplot(adult1, aes(x = diabetes_dx, y = count, fill = sex)) +
geom_bar(stat = "identity", position = "dodge") +
facet_wrap(~race) +
labs(title = "Diabetes Diagnosis by Sex and Race",
x = "Diabetes Diagnosis",
y = "Count") +
theme_minimal() +
scale_fill_manual(values = c("skyblue", "orange"))
Abnormalities detected in Adult dataset
Missingness
- Only 1.3% of individual data points are missing across the dataset, reflecting minimal missingness.
- No column is entirely missing (0%), indicating all variables have at least some observed data.
- Overall missingness: ~4% → low, but non-trivial given the small number of variables involved.
- Missingness is not completely at random (MNAR or MAR) - If the probability of missingness depends on other observed variables (e.g., older adults missing BMI due to illness), imputation helps reduce bias. It is possible and should consider MICE and test with logistic regression of missingness indicators
- Missingness affects outcome or key covariates - Even small missingness in important variables can bias posterior estimates. Since BMI and diabetes are central we should perform MICE
- Sufficient auxiliary variables available - MICE works best when you have other correlated variables to inform imputation (e.g., age, sex, race, WTMEC2YR).
- Bayesian model assumes complete data - Standard Bayesian logistic models (e.g., brms, rstanarm) cannot directly handle NAs — you must impute or model missingness.
Statistical Modeling
Multiple Logistic Regression model (Survey weighted Modeling for complete-case Analysis)
Code
# Modeling
library(broom)
library(mice)
library(brms)
library(posterior)
library(bayesplot)
library(knitr)
# --- Guardrails for modeling ---
n_outcome <- sum(!is.na(adult$diabetes_dx))
if (n_outcome == 0) stop("Too few non-missing outcomes for modeling. n = 0")
# Ensure factors and >=2 observed levels among complete outcomes
adult <- adult %>%
dplyr::mutate(
sex = if (!is.factor(sex)) factor(sex) else sex,
race = if (!is.factor(race)) factor(race) else race
)
if (nlevels(droplevels(adult$sex[!is.na(adult$diabetes_dx)])) < 2)
stop("sex has <2 observed levels after filtering; check data availability.")
if (nlevels(droplevels(adult$race[!is.na(adult$diabetes_dx)])) < 2)
stop("race has <2 observed levels after filtering; check Data Prep.")
# Survey-weighted complete-case
# Build a logical filter on the original adult data (same length as design$data)
keep_cc <- with(
adult,
!is.na(diabetes_dx) & !is.na(age_c) & !is.na(bmi_c) &
!is.na(sex) & !is.na(race)
)
# Subset the survey design using the logical vector (same length as original)
des_cc <- subset(nhanes_design_adult, keep_cc)
# Corresponding complete-case data (optional)
cc <- adult[keep_cc, ] |> droplevels()
cat("\nComplete-case N for survey-weighted model:", nrow(cc), "\n")
Complete-case N for survey-weighted model: 5349 Code
print(table(cc$race))
NH White Mexican American Other Hispanic NH Black
2293 713 470 1101
Other/Multi
772 Code
print(table(cc$diabetes_dx))
0 1
4752 597 Code
print(table(cc$sex))
Male Female
2551 2798 Code
form_cc <- diabetes_dx ~ age_c + bmi_c + sex + race
svy_fit <- survey::svyglm(formula = form_cc, design = des_cc, family = quasibinomial())
summary(svy_fit)
Call:
svyglm(formula = form_cc, design = des_cc, family = quasibinomial())
Survey design:
subset(nhanes_design_adult, keep_cc)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.67143 0.11935 -22.383 1.68e-08 ***
age_c 1.10833 0.05042 21.981 1.94e-08 ***
bmi_c 0.63412 0.05713 11.099 3.88e-06 ***
sexFemale -0.63844 0.10926 -5.843 0.000386 ***
raceMexican American 0.71091 0.13681 5.196 0.000826 ***
raceOther Hispanic 0.46469 0.13474 3.449 0.008712 **
raceNH Black 0.51221 0.15754 3.251 0.011677 *
raceOther/Multi 0.84460 0.17756 4.757 0.001433 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for quasibinomial family taken to be 0.8455444)
Number of Fisher Scoring iterations: 6- Strongest predictors: Age and BMI show the largest ORs among continuous variables—both strongly linked to diabetes risk.
- Protective factor: Being female reduces the odds of diabetes.
- Race disparities: All racial/ethnic minority groups show significantly higher odds compared to Non-Hispanic Whites, consistent with known health disparities in diabetes prevalence.
- Significance: All p-values < 0.05, so all predictors are statistically significant.
Handling Missing Data: Multivariate Imputation by Chained Equations (MICE)
- Variables with missing data are imputed conditionally on all others through iterative regression models.
- Multiple (m = 5–10) imputed datasets generated (MICE) are analyzed separately, and combined using Rubin’s rules to obtain pooled parameter estimates and standard errors.
Bayesian Logistic Regression
- Bayesian logistic regression model applied to the imputed datasets, with prior distributions incorporated and direct estimation of posterior distributions, credible intervals, and posterior predictive checks. Bayesian inference provided a probabilistic interpretation of parameter uncertainty, complementing the frequentist findings.
Model Validation and Interpretation
- Diagnostic checks performed below evaluate model convergence, goodness-of-fit, and predictive accuracy.
Model Comparison
- The results from both frameworks (frequentist and Bayesian) are compared to ensure robustness of conclusions regarding predictors of diabetes.
Code
svy_or <- broom::tidy(svy_fit, conf.int = TRUE) %>%
dplyr::mutate(OR = exp(estimate), LCL = exp(conf.low), UCL = exp(conf.high)) %>%
dplyr::select(term, OR, LCL, UCL, p.value) %>%
dplyr::filter(term != "(Intercept)")
knitr::kable(svy_or, caption = "Survey-weighted odds ratios (per 1 SD)")| term | OR | LCL | UCL | p.value |
|---|---|---|---|---|
| age_c | 3.0292807 | 2.6967690 | 3.4027912 | 0.0000000 |
| bmi_c | 1.8853571 | 1.6526296 | 2.1508579 | 0.0000039 |
| sexFemale | 0.5281132 | 0.4104905 | 0.6794397 | 0.0003857 |
| raceMexican American | 2.0358434 | 1.4850041 | 2.7910081 | 0.0008262 |
| raceOther Hispanic | 1.5915182 | 1.1664529 | 2.1714810 | 0.0087119 |
| raceNH Black | 1.6689718 | 1.1605895 | 2.4000450 | 0.0116773 |
| raceOther/Multi | 2.3270527 | 1.5451752 | 3.5045697 | 0.0014331 |
Multivariate Imputation by Chained Equations (Pooled Logistic Regression)
MICE, as an alternative to the Bayesian Approach Buuren and Groothuis-Oudshoorn (2011), manages missiging data.
Flatness of the density, heavy tails, non-zero peakedness, skewness and multimodality do not hamper performance of multiple imputation for the mean structure (n > 400) even for high percentages (75%) of missing data in one variable Van Buuren and Van Buuren (2012).
Multiple Imputation (MI) performed using mice R package, imputes missing values of one variable at a time, using regression models based on the other variables in the dataset.
In the chain process, each imputed variable become a predictor for the subsequent imputation, and the entire process is repeated multiple times to create several complete datasets, each reflecting different possibilities for the missing data.
Code
# ----- Multiple Imputation (predictors only)
mi_dat <- adult %>%
dplyr::select(diabetes_dx, age, bmi, sex, race, WTMEC2YR, SDMVPSU, SDMVSTRA)
meth <- mice::make.method(mi_dat)
pred <- mice::make.predictorMatrix(mi_dat)
# Do not impute outcome
meth["diabetes_dx"] <- ""
pred["diabetes_dx", ] <- 0
pred[,"diabetes_dx"] <- 1
# Imputation methods
meth["age"] <- "norm"
meth["bmi"] <- "pmm"
meth["sex"] <- "polyreg"
meth["race"] <- "polyreg"
# Survey design vars as auxiliaries only
meth[c("WTMEC2YR","SDMVPSU","SDMVSTRA")] <- ""
pred[, c("WTMEC2YR","SDMVPSU","SDMVSTRA")] <- 1
glimpse(mi_dat)Rows: 5,769
Columns: 8
$ diabetes_dx <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ age <dbl> 69, 54, 72, 73, 56, 61, 42, 56, 65, 26, 76, 33, 32, 38, 50…
$ bmi <dbl> 26.7, 28.6, 28.9, 19.7, 41.7, 35.7, NA, 26.5, 22.0, 20.3, …
$ sex <fct> Male, Male, Male, Female, Male, Female, Male, Female, Male…
$ race <fct> NH Black, NH White, NH White, NH White, Mexican American, …
$ WTMEC2YR <dbl> 13481.04, 24471.77, 57193.29, 65541.87, 25344.99, 61758.65…
$ SDMVPSU <dbl> 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2…
$ SDMVSTRA <dbl> 112, 108, 109, 116, 111, 114, 106, 112, 112, 113, 116, 114…Code
imp <- mice::mice(mi_dat, m = 5, method = meth, predictorMatrix = pred, seed = 123)
iter imp variable
1 1 bmi
1 2 bmi
1 3 bmi
1 4 bmi
1 5 bmi
2 1 bmi
2 2 bmi
2 3 bmi
2 4 bmi
2 5 bmi
3 1 bmi
3 2 bmi
3 3 bmi
3 4 bmi
3 5 bmi
4 1 bmi
4 2 bmi
4 3 bmi
4 4 bmi
4 5 bmi
5 1 bmi
5 2 bmi
5 3 bmi
5 4 bmi
5 5 bmiMICE Results:
After MICE, the final pooled imputed dataset consisted of 5,769 participants with 8 variables with missing values were addressed
Five imputations across five iterations each, with BMI imputed conditionally based on other predictors (age, sex, race, and diabetes status).
The iterative process showed stable convergence, indicating reliable estimation of missing BMI values for subsequent survey-weighted and Bayesian modeling analyses.
Code
fit_mi <- with(imp, {
age_c <- as.numeric(scale(age))
bmi_c <- as.numeric(scale(bmi))
glm(diabetes_dx ~ age_c + bmi_c + sex + race, family = binomial())
})
pool_mi <- pool(fit_mi)
summary(pool_mi) term estimate std.error statistic df p.value
1 (Intercept) -2.6895645 0.09941301 -27.054453 5566.204 1.486581e-151
2 age_c 1.0660265 0.05594733 19.054108 5520.446 1.911564e-78
3 bmi_c 0.5468538 0.04473386 12.224604 5148.557 6.751227e-34
4 sexFemale -0.6178297 0.09379129 -6.587282 5551.660 4.892566e-11
5 raceMexican American 0.8877355 0.13750463 6.456041 5472.583 1.167455e-10
6 raceOther Hispanic 0.5606621 0.17485537 3.206433 5573.987 1.351505e-03
7 raceNH Black 0.6809629 0.11981185 5.683602 5576.734 1.385727e-08
8 raceOther/Multi 0.7476406 0.15300663 4.886328 4749.963 1.061140e-06Code
## table
mi_or <- summary(pool_mi, conf.int = TRUE, exponentiate = TRUE) %>%
dplyr::rename(
term = term, OR = estimate, LCL = `2.5 %`, UCL = `97.5 %`, p.value = p.value
) %>%
dplyr::filter(term != "(Intercept)")
knitr::kable(mi_or, caption = "MI pooled odds ratios (per 1 SD)")| term | OR | std.error | statistic | df | p.value | LCL | UCL | conf.low | conf.high | |
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | age_c | 2.9038183 | 0.0559473 | 19.054108 | 5520.446 | 0.0000000 | 2.6021752 | 3.2404277 | 2.6021752 | 3.2404277 |
| 3 | bmi_c | 1.7278084 | 0.0447339 | 12.224604 | 5148.557 | 0.0000000 | 1.5827382 | 1.8861754 | 1.5827382 | 1.8861754 |
| 4 | sexFemale | 0.5391132 | 0.0937913 | -6.587282 | 5551.660 | 0.0000000 | 0.4485669 | 0.6479368 | 0.4485669 | 0.6479368 |
| 5 | raceMexican American | 2.4296216 | 0.1375046 | 6.456041 | 5472.583 | 0.0000000 | 1.8555327 | 3.1813298 | 1.8555327 | 3.1813298 |
| 6 | raceOther Hispanic | 1.7518320 | 0.1748554 | 3.206433 | 5573.987 | 0.0013515 | 1.2434346 | 2.4680953 | 1.2434346 | 2.4680953 |
| 7 | raceNH Black | 1.9757793 | 0.1198118 | 5.683602 | 5576.734 | 0.0000000 | 1.5621842 | 2.4988753 | 1.5621842 | 2.4988753 |
| 8 | raceOther/Multi | 2.1120110 | 0.1530066 | 4.886328 | 4749.963 | 0.0000011 | 1.5646727 | 2.8508138 | 1.5646727 | 2.8508138 |
Survey-weighted logistic regression coefficients (log-odds scale)
Regression coefficients for predictors of diabetes diagnosis (diabetes_dx) with the reference group (Male, Non-Hispanic White, average BMI and age) represents the change in log-odds of diabetes associated with a one-unit increase in the predictor (or compared to the reference group), controlling for all other variables.
Baseline log-odds of diabetes = -2.69
For each 1 SD increase in age, the log-odds of diabetes increase by 1.07 → odds increase by exp(1.07) = 2.9× (≈3× higher odds).
For each 1 SD increase in BMI, odds of diabetes increase by exp(0.55) = 1.73× (≈73% higher).
Females have exp(-0.62) = 0.54× the odds of diabetes compared to males → about 46% lower odds.
Mexican Americans have exp(0.89) = 2.43× higher odds of diabetes vs. Non-Hispanic Whites. Other Hispanics have exp(0.56) = 1.75× higher odds Non-Hispanic Blacks have exp(0.68) = 1.97× higher odds. Those identifying as “Other/Multi-racial” have exp(0.75) = 2.12× higher odds of diabetes.
Interpretation
Age and BMI are strong positive predictors of diabetes — each 1 SD increase in these variables substantially raises the odds.
Sex: Females show significantly lower odds compared to males.
Race/Ethnicity: All non-White racial groups have significantly higher odds of diabetes, highlighting persistent disparities in diabetes risk.
Model significance: All predictors are statistically significant (p < 0.01), suggesting a robust association across demographic and health variables.
Glimpse and statistics (Imputed dataset): - age (mean (SD)) = 48.84 (17.57) - females 2923 (52.3%) > males 2669 (47.7%) - majority being NH White count = 2398 (42.9%) - non-diabetics: Diabetics :: 4974 (88.9%): 618 (11.1%)
Code
library(gt)
# Bayesian Logistic Regression (formula weights)
adult_imp1 <- complete(imp, 1) %>%
dplyr::mutate(
age_c = as.numeric(scale(age)),
bmi_c = as.numeric(scale(bmi)),
wt_norm = WTMEC2YR / mean(WTMEC2YR, na.rm = TRUE),
# ensure factor refs match survey/mice:
race = forcats::fct_relevel(race, "NH White"),
sex = forcats::fct_relevel(sex, "Male")
) %>%
dplyr::filter(!is.na(diabetes_dx), !is.na(age_c), !is.na(bmi_c),
!is.na(sex), !is.na(race)) %>%
droplevels()
stopifnot(all(is.finite(adult_imp1$wt_norm)))
glimpse(adult_imp1)Rows: 5,592
Columns: 11
$ diabetes_dx <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ age <dbl> 69, 54, 72, 73, 56, 61, 42, 56, 65, 26, 76, 33, 32, 38, 50…
$ bmi <dbl> 26.7, 28.6, 28.9, 19.7, 41.7, 35.7, 23.6, 26.5, 22.0, 20.3…
$ sex <fct> Male, Male, Male, Female, Male, Female, Male, Female, Male…
$ race <fct> NH Black, NH White, NH White, NH White, Mexican American, …
$ WTMEC2YR <dbl> 13481.04, 24471.77, 57193.29, 65541.87, 25344.99, 61758.65…
$ SDMVPSU <dbl> 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2…
$ SDMVSTRA <dbl> 112, 108, 109, 116, 111, 114, 106, 112, 112, 113, 116, 114…
$ age_c <dbl> 1.13241831, 0.27835981, 1.30323001, 1.36016725, 0.39223428…
$ bmi_c <dbl> -0.33319172, -0.06755778, -0.02561558, -1.31184309, 1.7639…
$ wt_norm <dbl> 0.3393916, 0.6160884, 1.4398681, 1.6500477, 0.6380722, 1.5…Code
library(tableone)
vars <- c("age", "bmi", "age_c", "bmi_c", "wt_norm", "sex", "race", "diabetes_dx")
table1 <- CreateTableOne(vars = vars, data = adult_imp1, factorVars = c("sex", "race", "diabetes_dx"))
print(table1, showAllLevels = TRUE)
level Overall
n 5592
age (mean (SD)) 48.84 (17.57)
bmi (mean (SD)) 29.00 (7.11)
age_c (mean (SD)) -0.02 (1.00)
bmi_c (mean (SD)) -0.01 (0.99)
wt_norm (mean (SD)) 1.00 (0.79)
sex (%) Male 2669 (47.7)
Female 2923 (52.3)
race (%) NH White 2398 (42.9)
Mexican American 742 (13.3)
Other Hispanic 489 ( 8.7)
NH Black 1141 (20.4)
Other/Multi 822 (14.7)
diabetes_dx (%) 0 4974 (88.9)
1 618 (11.1) Code
## correlation matrix
library(ggplot2)
library(reshape2)
correlation_matrix <- cor(adult_imp1[, c("diabetes_dx", "age", "bmi")], use = "complete.obs", method = "pearson")
correlation_melted <- melt(correlation_matrix)
ggplot(correlation_melted, aes(Var1, Var2, fill = value)) +
geom_tile(color = "white") +
scale_fill_gradient2(low = "blue", high = "red", mid = "white", midpoint = 0,
limit = c(-1, 1), space = "Lab", name = "Correlation") +
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
labs(title = "Correlation Heatmap", x = "Features", y = "Features")
Visualization of the imputed dataset:
- Correlation matrix: Pairwise correlations heatmap: show the strength and direction of correlations (Pearson correlation) which measures linear association between diabetes_dx, age, and bmi
- Diabetes Diagnosis Distribution
- BMI Distribution by Diabetes Status
- Predicted Probability of Diabetes vs BMI
Code
# Class distribution
ggplot(adult_imp1, aes(x = factor(diabetes_dx))) +
geom_bar(fill = "steelblue") +
labs(
title = "Diabetes Diagnosis Distribution",
x = "Diabetes Diagnosis (0 = No, 1 = Yes)",
y = "Count"
) +
theme_minimal()
Code
prop.table(table(adult_imp1$diabetes_dx))
0 1
0.889485 0.110515 Code
# Visualization of Diabetes vs BMI (adult_data1)
library(ggplot2)
# Create the plot
ggplot(adult_imp1, aes(x = factor(diabetes_dx), y = bmi, fill = factor(diabetes_dx))) +
geom_boxplot(alpha = 0.7) +
scale_x_discrete(labels = c("0" = "No Diabetes", "1" = "Diabetes")) +
labs(
x = "Diabetes Diagnosis",
y = "BMI",
title = "BMI Distribution by Diabetes Status"
) +
theme_minimal() +
theme(legend.position = "none")
Code
# logistic regression curve
ggplot(adult_imp1, aes(x = bmi, y = diabetes_dx)) +
geom_point(aes(y = diabetes_dx), alpha = 0.2, position = position_jitter(height = 0.02)) +
geom_smooth(method = "glm", method.args = list(family = "binomial"), se = TRUE, color = "blue") +
labs(
x = "BMI",
y = "Probability of Diabetes",
title = "Predicted Probability of Diabetes vs BMI"
) +
theme_minimal()
Number of study population accross the three datasets
- Rows: 10175 and Columns: 10 (survey-weighted, merged data)
- Rows: 5,769 and Columns: 12 (filtered data, adult)
- Rows: 5,592 and Columns: 11 (imputed data, adult_imp1)
Bayesian Logistic Regression and Analysis
- Model Overview
- A Bayesian logistic regression model fitted on the first imputed dataset (adult_imp1) assessed predictors of diabetes diagnosis.
- Bayesian performed on imputed dataset (adult_imp1) with survey weights-Normalized MEC exam weights (wt_norm) with mean 1.00 (SD 0.79)
- No missing values, continuous variables are standardized, categorical variables are correctly re-leveled for reference categories and weights are available for inclusion in the likelihood to account for survey design.
- Prior Specification
- Intercept prior: student_t(3, 0, 10) — allowing heavy tails for flexibility in the intercept estimate. R. V. D. Schoot et al. (2013)
- Regression coefficients prior: normal(0, 2.5) — providing weakly informative regularization provide gentle regularization, constraining extreme values without overpowering the data R. van de Schoot et al. (2021)
- Model Estimation
- Using four Markov Chain Monte Carlo (MCMC) chains, each with 2000 iterations (50% warm-up), and an adaptive delta of 0.95 ensure good chain convergence and reduce divergent transitions.
- Posterior summaries represent the central tendency and uncertainty around the model parameters through credible intervals (CrI).
Bayesian Logistic Regression specifications
Family: bernoulli Links: mu = logit
Formula: diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race
Data: adult_imp1 (Number of observations: 5592) Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1; total post-warmup draws = 4000 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1).
Code
library(gt)
priors <- c(
set_prior("normal(0, 2.5)", class = "b"),
set_prior("student_t(3, 0, 10)", class = "Intercept")
)
bayes_fit <- brm(
formula = diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race,
data = adult_imp1,
family = bernoulli(link = "logit"),
prior = priors,
chains = 4, iter = 2000, seed = 123,
control = list(adapt_delta = 0.95),
refresh = 0 # quiet Stan output
)Running /opt/R/4.4.2/lib/R/bin/R CMD SHLIB foo.c
using C compiler: ‘gcc (GCC) 11.5.0 20240719 (Red Hat 11.5.0-2)’
gcc -I"/opt/R/4.4.2/lib/R/include" -DNDEBUG -I"/opt/R/4.4.2/lib/R/library/Rcpp/include/" -I"/opt/R/4.4.2/lib/R/library/RcppEigen/include/" -I"/opt/R/4.4.2/lib/R/library/RcppEigen/include/unsupported" -I"/opt/R/4.4.2/lib/R/library/BH/include" -I"/opt/R/4.4.2/lib/R/library/StanHeaders/include/src/" -I"/opt/R/4.4.2/lib/R/library/StanHeaders/include/" -I"/opt/R/4.4.2/lib/R/library/RcppParallel/include/" -I"/opt/R/4.4.2/lib/R/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/opt/R/4.4.2/lib/R/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/usr/local/include -fpic -g -O2 -c foo.c -o foo.o
In file included from /opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/Core:19,
from /opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/Dense:1,
from /opt/R/4.4.2/lib/R/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22,
from <command-line>:
/opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: cmath: No such file or directory
679 | #include <cmath>
| ^~~~~~~
compilation terminated.
make: *** [/opt/R/4.4.2/lib/R/etc/Makeconf:195: foo.o] Error 1Code
prior_summary(bayes_fit) prior class coef group resp dpar nlpar lb ub
normal(0, 2.5) b
normal(0, 2.5) b age_c
normal(0, 2.5) b bmi_c
normal(0, 2.5) b raceMexicanAmerican
normal(0, 2.5) b raceNHBlack
normal(0, 2.5) b raceOtherDMulti
normal(0, 2.5) b raceOtherHispanic
normal(0, 2.5) b sexFemale
student_t(3, 0, 10) Intercept
tag source
user
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
userCode
summary(bayes_fit) # Bayesian model summary Family: bernoulli
Links: mu = logit
Formula: diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race
Data: adult_imp1 (Number of observations: 5592)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -2.66 0.09 -2.84 -2.50 1.00 4187 3510
age_c 1.09 0.06 0.97 1.22 1.00 3012 3098
bmi_c 0.63 0.05 0.53 0.72 1.00 3472 3315
sexFemale -0.66 0.10 -0.86 -0.46 1.00 4003 3052
raceMexicanAmerican 0.69 0.18 0.35 1.04 1.00 3526 2843
raceOtherHispanic 0.43 0.24 -0.07 0.89 1.00 4058 3114
raceNHBlack 0.54 0.15 0.24 0.82 1.00 3597 3177
raceOtherDMulti 0.82 0.19 0.45 1.19 1.00 3763 3257
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Results and Visualization
Bayesian Logistic Regression Formula: diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race
- All Rhat ≈ 1.00 → model convergence is excellent
- ESS values are large (>1000), indicating stable posterior sampling and indicate model reliability.
- Baseline log-odds of diabetes for a male, non-Hispanic White individual with mean BMI and age → corresponds to very low probability of diabetes (~7%).
- For each 1 SD increase in age, the odds of diabetes are about 3 times higher, with high certainty (credible interval well above 0).
- Each 1 SD increase in BMI is associated with an 88% increase in odds of diabetes — a strong, significant effect.
- Females have 48% lower odds of diabetes than males, credible interval excludes 0 → strong evidence.
- Mexican Americans have about 2× higher odds of diabetes compared with Non-Hispanic Whites. Credible interval is entirely >0 → significant.
- Other Hispanics have slightly higher odds (≈1.5×), but the credible interval overlaps 0 → uncertain evidence
- Non-Hispanic Blacks have ~1.7× higher odds, with credible interval >0 → significant association.
- Individuals identifying as Other/Multi-racial have more than double the odds of diabetes, strong evidence.
Below plots shows Distributions for Coefficients
- Distributions for Coefficients from adult_imp1 data for age and bmi
- The density plots of standardized BMI and Age from the imputed dataset (adult_imp1) show approximately normal distributions centered near zero, consistent with z-score standardization confirming that both predictors were properly centered and scaled prior to Bayesian modeling, ensuring comparability and numerical stability during estimation.
- Prior Distributions for Coefficients
- Priors for regression coefficients were drawn from a Normal(0, 2.5) distribution, representing weakly informative assumptions centered at zero with moderate spread. The prior density plots for Age (per 1 SD) and BMI (per 1 SD) demonstrate symmetric bell-shaped distributions, indicating no strong bias toward positive or negative effects before observing data.
Code
library(ggplot2)
# adult_imp1 plot
# Convert to long format
adult_long <- adult_imp1 %>%
select(bmi_c, age_c) %>%
pivot_longer(cols = everything(), names_to = "Coefficient", values_to = "Value")
# Plot
ggplot(adult_long, aes(x = Value, fill = Coefficient)) +
geom_density(alpha = 0.5) +
theme_minimal() +
labs(title = "Distributions for Coefficients from adult_imp1 data",
x = "Coefficient Value", y = "Density") +
scale_fill_manual(values = c("bmi_c" = "skyblue", "age_c" = "orange"))
Code
## prior draws
prior_draws <- tibble(
term = rep(c("Age (per 1 SD)", "BMI (per 1 SD)"), each = 4000),
value = c(rnorm(4000, 0, 2.5), rnorm(4000, 0, 2.5))
)
## Plot (prior) (age and bmi)
ggplot(prior_draws, aes(x = value, fill = term)) +
geom_density(alpha = 0.5) +
theme_minimal() +
labs(title = "Prior Distributions for Coefficients",
x = "Coefficient Value", y = "Density") +
scale_fill_manual(values = c("skyblue", "orange"))
Predictive checking and validation of Bayesian model
- Posterior Summaries (mean, median, 95% credible intervals)
- Convergence diagnostics (R-hat, effective sample size)
- plots to visualizes posterior distributions with high uncertainty, narrow distributions indicating precise estimates.
- Posterior Odds Ratios provides interpretation of the model coefficients on a multiplicative scale with reference categories: NH White (race), Male (sex).
- Posterior Predictive Checks (PPC) assesses how the model reproduces observed data and validate model fit.
- Visualizations of generated simulated datasets compared with the observed data show density overlays for both mean and SD. There was no large discrepancies indicating potential misfit; there was good alignment suggesting reliable predictions.
- MCMC Convergence endures reliable posterior estimates.
- MCMC Trace plots show chains for each parameter over iterations.
- Well-mixed chains without trends indicate convergence and stable posterior estimates.
- Model Fit -provided details to quantify predictive performance.
- The proportion of variance explained by the model: R² = 0.13 (13%) shows predictors are relevant but other factors (e.g., genetics, lifestyle, environment) also contribute to outcome variability.
- Correlation and Parameter Relationships (Optional)
- Pairwise plots (mcmc_pairs, posterior) – explore correlations between parameters.
- Histograms or density plots mcmc_hist() or mcmc_areas() of specific parameters detects no collinearity or dependencies among predictors
Code
library(brms)
plot(bayes_fit) # Posterior distributions
Code
bayes_R2(bayes_fit) # Model fit Estimate Est.Error Q2.5 Q97.5
R2 0.1313342 0.01265055 0.1064607 0.156078Posterior Distributions (Left Panels)
- All distributions look smooth and unimodal → no multimodality, confirming stable posteriors.
- Each histogram represents the distribution of sampled coefficient values after convergence across all MCMC draws:
- b_raceOtherHispanic: The posterior peaks around 0.4–0.5, with some spread below 0 and above 1.→ Suggests a modestly positive association with diabetes risk, but some uncertainty (credible interval overlaps 0).
- b_raceNHBlack: Centered around 0.5–0.6, with a narrower, symmetric shape. → Indicates a consistent positive effect—NH Black participants have higher odds of diabetes, and uncertainty is low.
- b_raceOtherDMulti: Centered around 0.8–0.9, with slightly wider spread but entirely above 0. → Stronger evidence for increased odds of diabetes among Other/Multi-racial individuals.
Trace Plots (Right Panels)
- Each shows 4 MCMC chains (different colors) across 1000 iterations:
- The chains mix well and overlap substantially, without visible trends or drifts → indicates good convergence.
- The parameter values oscillate around stable means with no systematic pattern → confirms stationarity.
- Combined with Rhat ≈ 1 and high ESS from your summary, the trace plots visually validate posterior convergence and independence.
Bayesian 𝑅^2 (model fit statics):
- explains about 13% of the variability in diabetes status, with credible uncertainty bounds suggesting reasonable but modest explanatory power.
- Explains the expected proportion of variance explained, averaged over the posterior distribution of parameters.
Results from Posterior
- Below is the tabulated format for Bayesian posterior odds ratios (95% CrI) — reference: NH White (race), Male (sex)
- The Bayesian logistic regression model identified significant associations between demographic and anthropometric factors and diabetes diagnosis.
- Age a strong predictor: each standardized unit increase in age was associated with nearly threefold higher odds of diabetes (OR = 2.99; 95% CrI = 2.64–3.37).
- BMI showed a strong positive association (OR = 1.87; 95% CrI = 1.71–2.05), higher body mass substantially increased diabetes risk.
- Female sex had lower odds of diabetes compared to males (OR = 0.52; 95% CrI = 0.42–0.63).
- Compared with Non-Hispanic Whites (reference group), several racial/ethnic groups had higher odds:
- Mexican Americans (OR = 2.00; 95% CrI = 1.41–2.84)
- Non-Hispanic Blacks (OR = 1.71; 95% CrI = 1.28–2.27)
- Other/Multi-racial individuals (OR = 2.27; 95% CrI = 1.56–3.28)
- Other Hispanics showed a positive but non-significant association (OR = 1.54; 95% CrI = 0.93–2.43).
Code
# Posterior ORs (drop intercept, clean labels)
bayes_or <- posterior_summary(bayes_fit, pars = "^b_") %>%
as.data.frame() %>%
tibble::rownames_to_column("raw") %>%
dplyr::mutate(
term = gsub("^b_", "", raw),
term = gsub("race", "race:", term),
term = gsub("sex", "sex:", term),
term = gsub("OtherDMulti", "Other/Multi", term),
term = gsub("OtherHispanic", "Other Hispanic", term),
OR = exp(Estimate),
LCL = exp(Q2.5),
UCL = exp(Q97.5)
) %>%
dplyr::select(term, OR, LCL, UCL) %>%
dplyr::filter(term != "Intercept")
knitr::kable(
bayes_or %>%
dplyr::mutate(dplyr::across(c(OR,LCL,UCL), ~round(.x, 2))),
digits = 2,
caption = "Bayesian posterior odds ratios (95% CrI) — reference: NH White (race), Male (sex)"
)| term | OR | LCL | UCL |
|---|---|---|---|
| age_c | 2.99 | 2.64 | 3.37 |
| bmi_c | 1.87 | 1.71 | 2.05 |
| sex:Female | 0.52 | 0.42 | 0.63 |
| race:MexicanAmerican | 2.00 | 1.41 | 2.84 |
| race:Other Hispanic | 1.54 | 0.93 | 2.43 |
| race:NHBlack | 1.71 | 1.28 | 2.27 |
| race:Other/Multi | 2.27 | 1.56 | 3.28 |
Across all analytic methods—survey-weighted maximum likelihood estimation (MLE), multiple imputation with pooled estimates (MICE), and Bayesian regression—the associations between BMI, age, and diabetes diagnosis were consistent in direction and magnitude.
For BMI, the odds ratios ranged from 1.73 (95% CI: 1.58–1.89) in the MICE-pooled model to 1.89 (95% CI: 1.65–2.15) in the survey-weighted MLE model, and 1.87 (95% CrI: 1.71–2.05) in the Bayesian model.
For age, the estimated odds ratios were 2.90 (95% CI: 2.60–3.24) using MICE, 3.03 (95% CI: 2.70–3.40) from the survey-weighted MLE model, and 2.99 (95% CrI: 2.64–3.37) in the Bayesian analysis.
Code
# Results
#Build compact results table (BMI & Age only)
library(dplyr);
library(tidyr);
library(knitr);
library(stringr)
# pretty "OR (LCL–UCL)" string
fmt_or <- function(or, lcl, ucl, digits = 2) {
paste0(
formatC(or, format = "f", digits = digits), " (",
formatC(lcl, format = "f", digits = digits), "–",
formatC(ucl, format = "f", digits = digits), ")"
)
}
# guardrails: require these to exist from Modeling
stopifnot(exists("svy_or"), exists("mi_or"), exists("bayes_or"))
for (nm in c("svy_or","mi_or","bayes_or")) {
if (!all(c("term","OR","LCL","UCL") %in% names(get(nm)))) {
stop(nm, " must have columns: term, OR, LCL, UCL")
}
}
svy_tbl <- svy_or %>% mutate(Model = "Survey-weighted MLE")
mi_tbl <- mi_or %>% mutate(Model = "mice pooled")
bayes_tbl <- bayes_or %>% mutate(Model = "Bayesian")
all_tbl <- bind_rows(svy_tbl, mi_tbl, bayes_tbl) %>%
mutate(term = case_when(
str_detect(term, "bmi_c|\\bBMI\\b") ~ "BMI (per 1 SD)",
str_detect(term, "age_c|\\bAge\\b") ~ "Age (per 1 SD)",
TRUE ~ term
)) %>%
filter(term %in% c("BMI (per 1 SD)", "Age (per 1 SD)")) %>%
mutate(OR_CI = fmt_or(OR, LCL, UCL, digits = 2)) %>%
select(Model, term, OR_CI) %>%
arrange(
factor(Model, levels = c("Survey-weighted MLE","mice pooled","Bayesian")),
factor(term, levels = c("BMI (per 1 SD)","Age (per 1 SD)"))
)
res_wide <- all_tbl %>%
pivot_wider(names_from = term, values_from = OR_CI) %>%
rename(
`BMI (per 1 SD) OR (95% CI)` = `BMI (per 1 SD)`,
`Age (per 1 SD) OR (95% CI)` = `Age (per 1 SD)`
)
kable(
res_wide,
align = c("l","c","c"),
caption = "Odds ratios (per 1 SD) with 95% CIs across models"
)| Model | BMI (per 1 SD) OR (95% CI) | Age (per 1 SD) OR (95% CI) |
|---|---|---|
| Survey-weighted MLE | 1.89 (1.65–2.15) | 3.03 (2.70–3.40) |
| mice pooled | 1.73 (1.58–1.89) | 2.90 (2.60–3.24) |
| Bayesian | 1.87 (1.71–2.05) | 2.99 (2.64–3.37) |
Code
# Posterior predictive draws
#Posterior predictive checks (binary outcome)
pp_samples <- posterior_predict(bayes_fit, ndraws = 500) # 500 draws
# Check dimensions
dim(pp_samples) # rows = draws, cols = observations[1] 500 5592Comparative Visualizations (Predicted vs observed)
A total of 500 posterior predictive draws were generated for 5,592 observations, producing a simulated distribution of predicted outcomes consistent with the sample size. These draws were used to assess model fit and evaluate how well the Bayesian model reproduced the observed data pattern.
Code
# Plot overlay of observed vs predicted counts (duplicate image)
ppc_dens_overlay(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:50, ]) +
labs(title = "Posterior Predictive Check: Density Overlay") +
theme_minimal()
Posterior Predictive Check: Density Overlay
- model’s predictions align with reality where mean(y_rep) = average predicted probability of diabetes for each individual, across all posterior draws of the parameters. y = the actual observed diabetes status (0 = non-diabetic, 1 = diabetic).
- mcmc dens plots compare observed and posterior parameter values (estimates) for bmi_c, age_c, sex_female, and by race categories (1) Fitted (Predicted) vs observed for bmi using point and error bars (2) Fitted (Predicted) vs observed for bmi using line plot
Code
ppc_bars(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:50, ])
A posterior predictive check (Bar plot)
- using ppc_bars() compared the observed diabetes outcomes with 50 replicated datasets drawn from the posterior distribution. The replicated distributions closely matched the observed proportions, indicating that the Bayesian model adequately captured the outcome variability and overall data structureCode
#PP check for proportions (useful for binary) mean comparison to check if the simulated means match the observed mean
## mean
ppc_stat(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:100, ], stat = "mean") +
labs(title = "Posterior Predictive Check: Mean of Replicates") +
theme_minimal()
Posterior Predictive Check: Mean
-A posterior predictive check was performed on the mean diabetes outcome using 100 replicated datasets from the posterior distribution. The distribution of the simulated means closely aligned with the observed mean, suggesting that the Bayesian model accurately captures the central tendency of the outcome.Code
#PP check for proportions (useful for binary) mean and sd comparison to check if the simulated means match the observed mean
## sd
ppc_stat(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:100, ], stat = "sd") +
labs(title = "PPC: Standard Deviation of Replicates") +
theme_minimal()
Posterior Predictive Check: Standard Deviation
- A posterior predictive check was conducted on the standard deviation of diabetes outcomes using 100 replicated datasets from the posterior distribution. The simulated standard deviations closely matched the observed value, indicating that the Bayesian model adequately captures the variability in the outcome data.Code
library(brms)
library(dplyr)
library(posterior)
library(bayesplot)
library(ggplot2)
# Extract posterior draws as a draws_df # simulate posterior outcomes
post <- as_draws_df(bayes_fit)
post# A draws_df: 1000 iterations, 4 chains, and 11 variables
b_Intercept b_age_c b_bmi_c b_sexFemale b_raceMexicanAmerican
1 -2.6 1.1 0.70 -0.71 0.67
2 -2.7 1.0 0.62 -0.57 0.65
3 -2.6 1.1 0.65 -0.76 0.63
4 -2.7 1.0 0.65 -0.67 0.82
5 -2.6 1.1 0.61 -0.73 0.75
6 -2.5 1.0 0.60 -0.77 0.61
7 -2.8 1.1 0.66 -0.66 0.52
8 -2.8 1.2 0.67 -0.57 0.94
9 -2.8 1.1 0.65 -0.52 0.84
10 -2.6 1.1 0.67 -0.85 0.70
b_raceOtherHispanic b_raceNHBlack b_raceOtherDMulti
1 0.605 0.52 0.95
2 0.338 0.45 0.69
3 0.566 0.63 0.54
4 0.453 0.61 0.78
5 0.090 0.50 0.62
6 0.015 0.48 0.60
7 0.736 0.50 0.84
8 0.913 0.57 1.07
9 0.570 0.66 0.81
10 0.467 0.54 0.97
# ... with 3990 more draws, and 3 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}Code
# Posterior summary
post_sum <- posterior_summary(bayes_fit)
post_sum Estimate Est.Error Q2.5 Q97.5
b_Intercept -2.6644912 0.08841987 -2.840368e+00 -2.4963936
b_age_c 1.0936287 0.06218042 9.720147e-01 1.2162894
b_bmi_c 0.6267208 0.04755308 5.344481e-01 0.7198901
b_sexFemale -0.6586208 0.10156670 -8.592825e-01 -0.4575489
b_raceMexicanAmerican 0.6916992 0.17744231 3.465940e-01 1.0425831
b_raceOtherHispanic 0.4314438 0.24442275 -7.159507e-02 0.8867589
b_raceNHBlack 0.5379213 0.14668730 2.431286e-01 0.8182113
b_raceOtherDMulti 0.8190024 0.18868276 4.454199e-01 1.1877554
Intercept -2.6732989 0.06779455 -2.808774e+00 -2.5477114
lprior -16.5021561 0.05307875 -1.661688e+01 -16.4105283
lp__ -1430.3473284 2.03852832 -1.435302e+03 -1427.4169785Posterior Estimates for BMI and Age (posterior draws analysis)
The Bayesian model produced posterior estimates for the effects of BMI and age (standardized per 1 SD) on the outcome.
BMI showed a negative association in most draws, with posterior estimates ranging roughly from 0.61 to 0.70, indicating that higher BMI is associated with lower odds of the outcome in this analysis.
Age showed a positive association, with posterior estimates ranging roughly from 1.00 to 1.14, suggesting that higher age increases the odds of the outcome.
These posterior estimates reflect both the central tendency and variability in the effect of BMI and age, highlighting their roles as important predictors in the model.
Posterior summary
- The posterior summary of the Bayesian model reports the following for each parameter:
- Estimate: The posterior mean of the coefficient, representing the central tendency of the parameter’s distribution.
- Est.Error: The posterior standard deviation, quantifying uncertainty around the estimate.
- Q2.5: The 2.5th percentile of the posterior distribution, representing the lower bound of the 95% credible interval.
- Q97.5: The 97.5th percentile of the posterior distribution, representing the upper bound of the 95% credible interval.
Code
library(brms)
library(dplyr)
library(posterior)
library(bayesplot)
library(ggplot2)
# Density overlay for age and bmi
mcmc_areas(post, pars = c( "b_age_c","b_bmi_c","b_sexFemale","b_raceMexicanAmerican", "b_raceOtherHispanic","b_raceNHBlack","b_raceOtherDMulti" ))
Posterior Distributions of Model Coefficients
- Using 4,000 posterior draws from the Bayesian model (4 chains × 1,000 post-warmup draws per chain), the mcmc_areas() plot visualized the posterior distributions of key predictors: age, BMI, sex, and race/ethnicity.
- The posterior densities show the range and uncertainty of each coefficient.
- The 95% credible intervals are clearly depicted by the shaded areas, highlighting which predictors have strong evidence of association with diabetes.
- Age and BMI showed positive associations, female sex showed a negative association, and several racial/ethnic groups had elevated odds relative to the reference group.
- This visualization provides an intuitive overview of both the magnitude and uncertainty of the model’s estimated effects.
Code
predicted <- fitted(bayes_fit, summary = TRUE)
observed <- adult_imp1[, c("bmi", "age")]
# Plot for **bmi** (obs vs pred)
ggplot(data = NULL, aes(x = observed$bmi, y = predicted[, "Estimate"])) +
geom_point() +
geom_errorbar(aes(ymin = predicted[, "Q2.5"], ymax = predicted[, "Q97.5"])) +
geom_abline(slope = 1, intercept = 0, linetype = "dashed", color = "red") +
xlab("Observed bmi") + ylab("Predicted bmi")
Observed vs. Predicted BMI
- A comparison of observed BMI values with their posterior predicted estimates was performed using the Bayesian model.
- Each point represents an individual’s observed BMI versus the model’s predicted mean.
- Error bars indicate the 95% credible intervals of the predictions.
- The dashed red line represents perfect prediction (observed = predicted). - The plot demonstrates that the model’s predictions generally align with the observed data, with most points closely following the diagonal, indicating good predictive performance for BMI.
Code
library(posterior)
library(dplyr)
library(tidyr)
# Extract posterior draws as a matrix, then convert to tibble
post <- as_draws_matrix(bayes_fit) %>% # safer than as_draws_df for manipulation
as.data.frame() %>%
select(b_bmi_c, b_age_c) %>%
pivot_longer(
everything(),
names_to = "term",
values_to = "estimate"
) %>%
mutate(
term = case_when(
term == "b_bmi_c" ~ "BMI (per 1 SD)",
term == "b_age_c" ~ "Age (per 1 SD)"
),
type = "Posterior"
)
prior_draws <- tibble(
term = rep(c("BMI (per 1 SD)", "Age (per 1 SD)"), each = 4000),
estimate = c(rnorm(4000, 0, 1), rnorm(4000, 0, 1)),
type = "Prior"
)
combined_draws <- bind_rows(prior_draws, post)Prior vs Posterior Distributions
To assess how the Bayesian model updates beliefs from prior information to posterior estimates, we compared prior vs posterior coefficient distributions for key predictors: BMI and age. 1. Prior Draws - Simulated from a standard normal distribution (mean = 0, SD = 1) for both BMI and age coefficients. Represent initial beliefs about coefficient values before seeing the data. 2. Posterior Draws - Extracted from the fitted model (bayes_fit) for b_bmi_c and b_age_c. - Pivoted to long format and labeled as “Posterior”. 3. Visualization Combined prior and posterior draws - Plotted density overlays with facets for BMI and age. - Posterior distributions are narrower and often shifted from prior, reflecting information gained from the data. - Differences between prior and posterior highlight the model’s learning about effect sizes. - Posterior Predictive Proportions of Diabetes - Computed the proportion of diabetes cases (diabetes = 1) for each posterior draw (pp_samples).
Interpretaion: - Prior vs posterior plots demonstrate that the Bayesian model updates prior beliefs in a data-informed way. - Posterior predictive proportions closely match observed prevalence, supporting model reliability for inference and prediction.
Code
library(ggplot2)
ggplot(combined_draws, aes(x = estimate, fill = type)) +
geom_density(alpha = 0.4) +
facet_wrap(~ term, scales = "free", ncol = 2) +
theme_minimal(base_size = 13) +
labs(
title = "Prior vs Posterior Distributions",
x = "Coefficient estimate",
y = "Density",
fill = ""
)
Prior vs. Posterior Distributions - We visualized the prior and posterior distributions of the BMI and age coefficients. - Priors were centered at 0, reflecting weak prior beliefs about the direction and magnitude of the effects. - Posteriors were shifted away from 0, indicating that the data provided strong evidence for associations with the outcome. - The density plots highlight the uncertainty and magnitude of the estimated effects, with posterior distributions narrower than the priors, demonstrating that the data meaningfully updated our beliefs. - Faceting by term allows comparison of each predictor’s prior and posterior distributions.
Code
# Compute proportion of diabetes=1 for each draw
pp_proportion <- rowMeans(pp_samples) # proportion of 1's in each posterior draw
# Optional: visualize the posterior probability distribution
pp_proportion_df <- tibble(proportion = pp_proportion)
ggplot(pp_proportion_df, aes(x = proportion)) +
geom_histogram(binwidth = 0.01, fill = "skyblue", color = "black") +
labs(
title = "Posterior Distribution of Proportion of Diabetes = 1",
x = "Proportion of Diabetes = 1",
y = "Frequency"
) +
theme_minimal()
Posterior Distribution of Diabetes Prevalence
- A histogram of these values shows the distribution of predicted prevalence calculated for each draw of the Bayesian model (pp_proportion), reflecting uncertainty in the model estimates, central tendency and variability of diabetes prevalence
- Most posterior predictions cluster around 10–11%, indicating good alignment with the observed/imputed data and demonstrating that the model captures the underlying population pattern.
Code
library(dplyr)
library(knitr)
svy_mean <- svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)
# Create summary table
summary_table <- tibble(
Method = c("Survey-weighted mean (NHANES)",
"Imputed dataset mean",
"Posterior predictive mean"),
diabetes_mean = c(
coef(svy_mean), # survey-weighted mean
mean(adult_imp1$diabetes_dx, na.rm = TRUE), # imputed dataset
mean(pp_proportion) # posterior predictive mean
),
SE = c(
SE(svy_mean), # survey-weighted SE
NA, # not available for raw mean
NA # not available for posterior predictive
)
)
# Render table
kable(summary_table, digits = 4, caption = "Comparison of Diabetes Prevalence Across Methods")| Method | diabetes_mean | SE |
|---|---|---|
| Survey-weighted mean (NHANES) | 0.0890 | 0.0048 |
| Imputed dataset mean | 0.1105 | NA |
| Posterior predictive mean | 0.1089 | NA |
Comparison of Diabetes Prevalence Across Methods
- The estimated prevalence of diabetes was consistent across different analytical approaches:
- Survey-weighted mean (NHANES): 8.9% (SE = 0.0048)
- Imputed dataset mean (MICE): 11.1%
- Posterior predictive mean (Bayesian model): 10.95%
These results indicate that multiple imputation and Bayesian posterior predictions yield slightly higher prevalence estimates than the raw survey-weighted mean, but all methods are broadly consistent. The posterior predictive distribution closely matches the observed prevalence, suggesting that the Bayesian model is well-calibrated.
Bayesian model predicts that about 10–11% of this population has diabetes, with a relatively narrow range across posterior draws, reflects uncertainty in the estimate
While most predictions cluster around 10–11%, the model allows for values as low as 8.5% and as high as 12.8%.
On comparing this with the raw imputed data proportion show that the the model predictions align with the observed/imputed data.
The predicted proportion incorporates uncertainty from both the Bayesian model and the imputed data, providing a more robust estimate of diabetes prevalence.
These results suggest that approximately 1 in 10 adults in this population may have diabetes, which can help policymakers and clinicians plan and prioritize targeted interventions effectively.
Code
library(tidyverse)
# Posterior predicted proportion vector
# pp_proportion <- rowMeans(pp_samples) # if not already done
known_prev <- 0.089 # NHANES prevalence
# Posterior summary
posterior_mean <- mean(pp_proportion)
posterior_ci <- quantile(pp_proportion, c(0.025, 0.975)) # 95% credible interval
# Create a data frame for plotting
pp_df <- tibble(proportion = pp_proportion)
# Plot
ggplot(pp_df, aes(x = proportion)) +
geom_histogram(binwidth = 0.005, fill = "skyblue", color = "black") +
geom_vline(xintercept = known_prev, color = "red", linetype = "dashed", size = 1) +
geom_vline(xintercept = posterior_mean, color = "blue", linetype = "solid", size = 1) +
geom_rect(aes(xmin = posterior_ci[1], xmax = posterior_ci[2], ymin = 0, ymax = Inf),
fill = "blue", alpha = 0.1, inherit.aes = FALSE) +
labs(
title = "Posterior Predicted Diabetes Proportion vs NHANES Prevalence",
subtitle = paste0("Red dashed = NHANES prevalence (", known_prev,
"), Blue solid = Posterior mean (", round(posterior_mean,3), ")"),
x = "Proportion of Diabetes = 1",
y = "Frequency"
) +
theme_minimal()
Posterior Predicted Diabetes Proportion vs. NHANES Prevalence
- We compared the posterior predicted proportion of diabetes from the Bayesian model with the observed NHANES prevalence (8.9%).
- The blue solid line represents the posterior mean, while the shaded blue area indicates the 95% credible interval of predicted proportions.
- The red dashed line shows the NHANES survey-weighted prevalence for reference.
- Most posterior predictions cluster around 10–11%, slightly higher than the NHANES mean, but the credible interval overlaps the observed prevalence, indicating good model calibration.
- This visualization highlights the model’s ability to capture uncertainty in predictions while remaining consistent with the observed data.
Code
library(dplyr)
# Posterior predicted proportion
posterior_mean <- mean(pp_proportion)
posterior_ci <- quantile(pp_proportion, c(0.025, 0.975)) # 95% credible interval
# NHANES prevalence with SE from survey::svymean
# Suppose you already have:
# svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)
known_prev <- 0.089 # Mean prevalence
known_se <- 0.0048 # Standard error from survey
# Calculate 95% confidence interval
known_ci <- c(
known_prev - 1.96 * known_se,
known_prev + 1.96 * known_se
)
# Print results
data.frame(
Type = c("Posterior Prediction", "NHANES Prevalence"),
Mean = c(posterior_mean, known_prev),
Lower_95 = c(posterior_ci[1], known_ci[1]),
Upper_95 = c(posterior_ci[2], known_ci[2])
) Type Mean Lower_95 Upper_95
2.5% Posterior Prediction 0.1089181 0.09629381 0.1216962
NHANES Prevalence 0.0890000 0.07959200 0.0984080Code
# Create a data frame for plotting
ci_df <- data.frame(
Type = c("Posterior Prediction", "NHANES Prevalence"),
Mean = c(0.1096674, 0.089),
Lower_95 = c(0.09772443, 0.079592),
Upper_95 = c(0.1210658, 0.098408)
)
# --- 1. Survey-weighted (Population) prevalence ---
pop_est <- svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)
pop_prev <- as.numeric(pop_est)
pop_se <- as.numeric(SE(pop_est))
pop_ci <- c(pop_prev - 1.96 * pop_se, pop_prev + 1.96 * pop_se)
# --- 2. Bayesian posterior prevalence ---
# bayes_pred = matrix of posterior draws (iterations × individuals)
pp_proportion <- rowMeans(pp_samples) # prevalence per posterior draw
post_prev <- mean(pp_proportion) # posterior mean prevalence
post_ci <- quantile(pp_proportion, c(0.025, 0.975)) # 95% credible interval
# --- 3. Combine into one data frame ---
bar_df <- tibble(
Source = c("Survey-weighted (Population)", "Bayesian Posterior"),
Prevalence = c(pop_prev, post_prev),
CI_low = c(pop_ci[1], post_ci[1]),
CI_high = c(pop_ci[2], post_ci[2])
)
# --- 4. Plot ---
ggplot(bar_df, aes(x = Source, y = Prevalence, fill = Source)) +
geom_col(alpha = 0.85, width = 0.6) +
geom_errorbar(
aes(ymin = CI_low, ymax = CI_high),
width = 0.15,
color = "black",
linewidth = 0.8
) +
guides(fill = "none") +
labs(
title = "Population vs Posterior Diabetes Prevalence",
subtitle = "Survey-weighted estimate (design-based) vs Bayesian (model-based)",
y = "Prevalence (Proportion with Diabetes)",
x = NULL
) +
theme_minimal(base_size = 13)
Comparison of Diabetes Prevalence Across Methods
- The bar plot compares survey-weighted, imputed, and Bayesian posterior estimates of diabetes prevalence:
- Survey-weighted prevalence: 8.9% (95% CI: 0.080–0.098), reflecting the NHANES population after accounting for complex sampling.
- Imputed (unweighted) prevalence: 11.1%, slightly higher due to unadjusted overrepresentation of subgroups with higher diabetes rates.
- Bayesian posterior mean: 10.9% (95% CrI: 0.098–0.121), closely replicating the imputed data while slightly shrinking toward the population-level mean, consistent with Bayesian regularization.
- The posterior credible interval overlaps the survey 95% CI, indicating that the Bayesian model reproduces population-level prevalence accurately. This demonstrates good model calibration and predictive validity, while visualizing the uncertainty of both survey-based and model-based estimates.
Practical Implications - Health departments can estimate diabetes burden at the state or county level using Bayesian small-area estimation. - Clinicians and public health researchers can plan targeted screening where predicted prevalence is higher than observed. - Epidemiologists can validate disease models before applying them to regions without survey data.
Code
library(tidyr)
library(bayesplot)
library(posterior)
# Convert fitted model to draws array
post_array <- as_draws_array(bayes_fit) # draws x chains x parameters
# Plot autocorrelation for age and bmi
mcmc_acf(post_array, pars = c("b_age_c", "b_bmi_c"))
MCMC Autocorrelation for Key Parameters of Posterior Samples
- Autocorrelation plots were generated for the posterior draws of age and BMI coefficients to assess chain mixing and convergence:
- The plots show the correlation of each draw with its lagged values across iterations.
- Rapid decay of autocorrelation toward zero indicates that the Markov chains are mixing well and successive draws are relatively independent.
- Both age and BMI coefficients exhibited low autocorrelation after a few lags, supporting the reliability of posterior estimates.
- This diagnostic confirms that the Bayesian model sampling was adequate and stable, ensuring valid inference from the posterior distributions.
Conclusion
Model Performance and Fit
- Across multiple modeling approaches — survey-weighted MLE, multiple imputation (MICE), and Bayesian logistic regression — the models were consistent and reliable in predicting diabetes risk.
- The Bayesian model showed excellent convergence (Rhat = 1.00; Bulk/Tail ESS > 2000) and explained approximately 13% of the variance in diabetes (Posterior R² = 0.13, 95% CrI: 0.106–0.156).
- Posterior predictive checks confirmed good model calibration, with credible intervals capturing predictive uncertainty.
Key Predictors
- Age and BMI were the strongest and most consistent predictors across all methods. Each 1 SD increase in age nearly tripled the odds of diabetes, while a 1 SD increase in BMI increased the odds by approximately 1.7–1.9×.
- Sex (female) was associated with lower odds of diabetes.
- Race/Ethnicity: Mexican American, NH Black, and Other/Multi groups had significantly higher odds; the effect for Other Hispanic was less certain.
Interpretation and Robustness
- Bayesian credible intervals were slightly narrower than frequentist confidence intervals but largely overlapped, indicating robust effect estimates and stable inference.
- Prior regularization in the Bayesian model stabilized estimates in the presence of modest missingness.
- The results are slightly conservative relative to the observed population prevalence, but the posterior predictions remain consistent with both survey-weighted and imputed data.
Design Considerations
- Survey-weighted MLE accounts for the complex NHANES sampling design, producing population-representative estimates.
- The Bayesian model used normalized weights as importance weights, which approximates the effect of survey weights but does not fully account for stratification, clustering, or design-based variance adjustments.
Overall: - Age and BMI are consistently strong risk factors for diabetes in this population. - The Bayesian model complements frequentist approaches by providing stable, interpretable, and uncertainty-quantified estimates, while broadly reproducing population-level prevalence.
Discussions
- The use of multiple imputation allowed for robust analysis despite missing data, increasing power and reducing bias.
- Comparison of frequentist and Bayesian models demonstrated consistency in significant predictors, while Bayesian approaches provided the advantage of posterior distributions and probabilistic interpretation.
- Across all models, both age and BMI emerged as strong and consistent predictors of diabetes.
- The consistency across modeling approaches strengthens the validity of these findings Multiple imputation accounted for potential biases due to missing data, and Bayesian modeling provided robust credible intervals that closely matched frequentist estimates align with previous epidemiological research indicating that increasing age and higher BMI are among the most important determinants of type 2 diabetes risk.
- Cumulative exposure to metabolic and lifestyle risk factors over time, and the role of excess adiposity and insulin related effects account for diabetes.
- Survey weighted dataset strenghthens ensuring population representativeness, multiple imputation to handle missing data, and rigorous Bayesian estimation provided high effective sample sizes and R̂ ≈ 1.00 across parameters confirmed excellent model convergence.
- Bayesian logistic regression provided inference statistically consistent and interpretable achieving the aim of this study. In future research hierarchical model using NHANES cycles or adding variables (lab tests) could assess nonlinear effects of metabolic risk factors.
Limitations
- The study is based on cross-sectional/observational NHANES data, which limits the ability to make causal inferences. Associations observed between BMI, age, diabetes status cannot confirm causation.
- The project relies on multiple imputation for missing values, even though imputation reduces bias, it assumes missingness is at random (MAR); if data are missing not at random (MNAR), results may be biased.
- Potential Residual Confounding - Models included key predictors (age, BMI, sex, race), but unmeasured factors like diet, physical activity, socioeconomic status, or genetic predisposition could confound associations.
- Bayesian models depend on prior choices, which could influence posterior estimates if priors are informative or mis-specified.
- Variable Measurement - BMI is measured at a single time point, which may not reflect long-term exposure or risk.
- Self-reported variables - are subjective to recall or reporting bias.
- Interactions are not tested in the study (bmi × age) and so other potential synergistic effects might be missed.
- Predicted probabilities are model-based estimates, not validated for clinical decision-making. External validation in independent cohorts is needed before application.
Targeted Therapy
- Translational Perspective from the Bayesian Diabetes Prediction Project. This project further demonstrates the translational potential of Bayesian modeling in clinical decision-making and public health strategy.
- By using patient-level predictors such as age, BMI, sex, and race to estimate the probability of diabetes, the model moves beyond descriptive statistics toward individualized risk prediction.
- The translational move lies in converting these probabilistic outputs into actionable thresholds—such as identifying the BMI or age at which the predicted risk of diabetes exceeds a clinically meaningful level (e.g., 30%).
- Such insights can guide early screening, personalized lifestyle interventions, and targeted prevention programs for populations at higher risk.
- This approach embodies precision public health—bridging data science and medical decision-making to deliver tailored, evidence-based strategies that can ultimately improve diabetes prevention and management outcomes.
What changes in modifiable predictors would lower diabetes risk?
Translational Research Implications:
- We can use the model to guide prevention or intervention.
- Only BMI is a modifiable risk factor
- We can make changes in BMI (behavior or lifestyle) to achieve a lower risk threshold
- we hold non modifiable predictors as constant (sex, race).
- Vary modifiable predictors (BMI) until the model predicts the desired probability.
Internal validation
- To illustrate personalized risk estimation using the Bayesian model, we computed the posterior predicted probability of diabetes for a representative participant.
- We selected one participant from the dataset (adult[1, ]) including all relevant covariates (age, BMI, sex, race).
- Used posterior_linpred with transform = TRUE to obtain predicted probabilities for logistic regression.
- Extracted posterior draws computed 95% credible interval from the posterior draws.
- Density plot shows the distribution of plausible probabilities given the participant’s covariates.
- The density highlights uncertainty around the individual’s predicted diabetes risk.
- 95% credible interval provides a range of probable outcomes, not just a point estimate.
- This approach allows personalized risk assessment, enabling clinicians or public health practitioners to identify high-risk individuals
- Tailor preventive interventions (e.g., lifestyle modification, monitoring)
- Quantify uncertainty in predictions for decision-making
- Posterior predictive distributions enable probabilistic, individualized predictions, supporting targeted intervention strategies beyond population-level summaries.
Code
# Use the first participant
# using multiple covariates to select someone
participant1_data <- adult[1, ]
# predicted probabilities for patient 1
phat1 <- posterior_linpred(bayes_fit, newdata = participant1_data, transform = TRUE)
# 'transform = TRUE' gives probabilities for logistic regression
# Store in a data frame for plotting
post_pred_df <- data.frame(pred = phat1)
# Compute 95% credible interval
ci_95_participant1 <- quantile(phat1, c(0.025, 0.975))
# Plot
ggplot(post_pred_df, aes(x = pred)) +
geom_density(color='darkblue', fill='lightblue') +
geom_vline(xintercept = ci_95_participant1[1], color='red', linetype='dashed') +
geom_vline(xintercept = ci_95_participant1[2], color='red', linetype='dashed') +
xlab('Probability of being diabetic (Outcome=1)') +
ggtitle('Posterior Predictive Distribution 95% Credible Interval') +
theme_bw()
Code
participant2_data <- adult[2, ]
# predicted probabilities for patient 1
phat2 <- posterior_linpred(bayes_fit, newdata = participant2_data, transform = TRUE)
# 'transform = TRUE' gives probabilities for logistic regression
# Store in a data frame for plotting
post_pred_df2 <- data.frame(pred = phat2)
# Compute 95% credible interval
ci_95_participant2 <- quantile(phat2, c(0.025, 0.975))
# Plot
ggplot(post_pred_df2, aes(x = pred)) +
geom_density(color='darkblue', fill='lightblue') +
geom_vline(xintercept = ci_95_participant2[1], color='red', linetype='dashed') +
geom_vline(xintercept = ci_95_participant2[2], color='red', linetype='dashed') +
xlab('Probability of being diabetic (Outcome=1)') +
ggtitle('Posterior Predictive Distribution 95% Credible Interval') +
theme_bw()
External validation
- Predicting Diabetes Risk for a New Participant to demonstrate the application of the Bayesian model for personalized prediction, we applied the trained model to a new participant not included in the original dataset.
- Selected a new participant with specific covariates (age, BMI, sex, race).
- Used posterior_linpred with transform = TRUE to compute posterior predicted probabilities of diabetes. Generated posterior draws to capture predictive uncertainty.
- Created a density plot of predicted probabilities. Computed 95% credible interval to summarize the range of likely outcomes.
- Red dashed lines indicate the lower and upper bounds of the interval.
- The distribution shows not only the most probable risk but also the uncertainty around it.
- Credible intervals help quantify confidence in individual-level predictions.
- Supports personalized decision-making, such as targeted lifestyle interventions, early monitoring, or preventive care.
- Bayesian posterior predictive draws allow probabilistic, individualized predictions for new participants, providing both point estimates and uncertainty measures for actionable risk assessment.
Code
library(ggplot2)
new_participant <- data.frame(
age_c = 40,
bmi_c = 25,
sex = "Female",
race = "Mexican American"
)
# Posterior predicted probabilities
phat_new <- posterior_linpred(bayes_fit, newdata = new_participant, transform = TRUE)
# Convert to numeric vector
phat_vec <- as.numeric(phat_new)
# Check the range to see if all values are similar
range(phat_vec)[1] 1 1Code
# Store in a data frame
post_pred_df_new <- data.frame(pred = phat_vec)
# Compute 95% credible interval from the vector
ci_95_new_participant <- quantile(phat_vec, c(0.025, 0.975))
# Plot
ggplot(post_pred_df_new, aes(x = pred)) +
geom_density(color='darkblue', fill='lightblue', alpha = 0.6) +
geom_vline(xintercept = ci_95_new_participant[1], color='red', linetype='dashed') +
geom_vline(xintercept = ci_95_new_participant[2], color='red', linetype='dashed') +
xlim(0, 1) + # ensures you see the curve even if values are close
xlab('Probability of being diabetic (Outcome=1)') +
ggtitle('Posterior Predictive Distribution (95% Credible Interval)') +
theme_bw()
To estimate Targeted BMI for Predicted Diabetes Risk
- To analyze the relationship between BMI and the predicted probability of diabetes, holding other covariates (age, sex, race) constant, via fitted Bayesian logistic regression model, we generated a grid of BMI values (e.g., 18–40 kg/m²) for a specific demographic profile: Age = 40 Sex = Female Race = Mexican American
- We computed posterior predicted probabilities of diabetes for each BMI value.
- Averaged across posterior draws to obtain the mean predicted probability per BMI.
- Target Probability Approach Defined a target probability of diabetes (e.g., 0.3). Identified the BMI value whose predicted probability is closest to the target. This enables inverse prediction, linking statistical inference to clinically meaningful thresholds.
- Visualization Line plot of predicted probability vs BMI shows
- The blue curve shows how likely different probability values are according to your posterior predictive distribution.-
- The red dashed lines show the 95% credible interval (CI) for this participant’s probability of being diabetic.
- Limits x-axis between 0 and 1 (valid range for probabilities).
- Red dashed horizontal line: target probability (0.3).
- Red dotted vertical line: BMI corresponding to the target probability (~closest BMI).Annotated to highlight the BMI threshold.
- The blue curve shows how likely different probability values are according to your posterior predictive distribution.-
- Provides a practical guideline:
- BMI at which an individual with a given profile reaches a predefined diabetes risk.
- Supports personalized risk communication and preventive interventions.
- Translates model output into actionable, clinically relevant thresholds, bridging research findings with public health application.
- This approach demonstrates how Bayesian posterior predictions can be used for targeted, individualized risk assessment, informing precision prevention strategies based on modifiable risk factors like BMI.
Practical Implications
- age and BMI as robust and independent predictors of diabetes, underscore the importance of early targeted interventions in mitigating diabetes risk.
- Longitudinal studies and combining other statistical analytical methods with Bayesian can further enhance and provide better informed precision prevention strategies.
Code
# Grid of possible BMI values (centered if model used bmi_c)
bmi_seq <- seq(18, 40, by = 0.5)
newdata_grid <- data.frame(
age_c = 40,
bmi_c = bmi_seq,
sex = "Female",
race = "Mexican American"
)
# Posterior mean predicted probabilities
pred_probs <- posterior_linpred(bayes_fit, newdata = newdata_grid, transform = TRUE)
# Average over posterior draws to get the mean predicted probability per BMI
prob_mean <- colMeans(pred_probs)
# Combine with BMI values
pred_df <- cbind(newdata_grid, prob_mean)
target_prob <- 0.3
closest <- pred_df[which.min(abs(pred_df$prob_mean - target_prob)), ]
closest age_c bmi_c sex race prob_mean
1 40 18 Female Mexican American 1Code
ggplot(pred_df, aes(x = bmi_c, y = prob_mean)) +
geom_line(color = "darkblue", linewidth = 1.2) +
geom_hline(yintercept = target_prob, color = "red", linetype = "dashed") +
geom_vline(xintercept = closest$bmi_c, color = "red", linetype = "dotted") +
annotate("text", x = closest$bmi_c, y = target_prob + 0.05,
label = paste0("Target BMI ≈ ", round(closest$bmi_c, 1)),
color = "red", hjust = -0.1) +
labs(
x = "BMI (centered or raw, depending on model)",
y = "Predicted Probability of Diabetes",
title = "Inverse Prediction: BMI Needed for Target Diabetes Risk"
) +
theme_bw()
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